Page 355 - Caldera Volcanism Analysis, Modelling and Response

330 Agust Gudmundsson

In contrast to a newly drilled borehole, a magma chamber is a long-lived fluid-filled
structure that is presumably in lithostatic equilibrium with its surroundings for most
of its lifetime. Thus, in Equation (1), the normal condition everywhere along the
margin of the chamber is p ¼ s 3 ð¼ s 1 Þ. During short periods, that is unrest
l
periods, the situation may be p 40, either because of added volume of magma
e
(absolute increase in the chamber excess pressure) or reduction in s 3 . It is during
these periods that the condition of Equation (1) may be satisfied, resulting in
rupture and sheet or dyke initiation.
The initiated sheet may propagate for only a short distance from the chamber,
and then become arrested (Gudmundsson, 2002; Gudmundsson and Brenner,
2005). Alternatively, the sheet may propagate to the surface and supply magma
to an eruption. Consider a chamber located in a host rock that behaves as
elastic and a sheet inclined (dipping) at an angle of a to the surface (Figure 13),
where the vertical co-ordinate z is positive upwards. Then it follows from the
Navier–Stokes equation (Lamb, 1932; Milne-Thompson, 1996) that the volumetric
flow rate Q e through an ideal, parallel-wall sheet (a magma-filled fracture) is
L
(Gudmundsson and Brenner, 2005):

3
Du W @p
e e
Q ¼ ðr r Þg sin a (2)
L r m
12m @L
e
In the symbol for the volumetric flow rate Q , the superscript e is used to indicate
L
that the fracture occurs in an elastic rock and the subscript L denotes the dip-
dimension (distance of magma transport) of the sheet. Here, Du is the aperture
(opening) of the fracture (similar to the thickness of the inclined sheet after

Figure 13 Flow of magma through a dyke or an inclined sheet to the surface of a caldera
(cf. Figures 5 and 8; modi¢ed from Gudmundsson and Brenner, 2005).The volumetric £ow rate
Q depends on the dip and dip dimension of the feeder. Avertical dyke has a dip dimension z.
An inclined sheet with a dip a has a dip dimension L. For the dyke and the sheet, the surface
trace length (outcrop length) perpendicular to the magma-£ow direction is denoted byW.The
excess magmatic pressure in the chamber is p e , and the di¡erence is depth between the point
of initiation of the vertical and the inclined sheet is denoted by X (cf. Equation (2)).